Existence of photon region for stationary axisymmetric black holes

Abstract: The black hole shadow is fundamentally connected to the structure of light rings and photon region in the background geometry. We investigate spherical photon orbits (SPOs) and photon regions in generic asymptotically flat, stationary, axisymmetric black hole spacetimes. We formally define the photon region's boundary purely in terms of the background metric functions, independent of the photon's parameters like energy or angular momentum, and establish the existence of at least one photon region for such spacetimes. We further analyze its common features including overlap with the ergoregion and rotation sense of SPOs. Additionally, light rings are identified at the extrema of the photon region boundary curves in the $(r,\theta)$ plane. Our approach is validated against a few exact black hole solutions. Implications are discussed.